We present an iterative path integral algorithm for computing multitime correlation functions of a quantum system coupled to a dissipative bath of harmonic oscillators. By splitting the Boltzmann operator into two parts and reordering the propagators in the expression for canonical correlation functions, we are able to transform the evolution time contour into a symmetric one so that a forward propagation and a backward one are specified. Because the memory induced by the bath through the Feynman-Vernon influence functional decays rapidly in the complex time plane, long-time correlations are negligible. Taking advantage of this fact, we show that the correlation function can be obtained via an iterative procedure. The method is used to calculate three-time correlation functions of a dissipative two-level system.